On the Cluster Property Above the Critical ... - Project Euclid

Commun. math. Phys. 12, 269—274 (1969)

On the Cluster Property Above the Critical Temperature in Lattice Gases G. GALLAVOTTI and S. MIRACLE-SOLE* Institut des Hautes Etudes Scientifiques 91 Bures-sur-Yvette — France Received January 2, 1969 Abstract. We prove cluster properties of the correlation functions at high temperature and arbitrary activity. We obtain also results on clustering at complex temperatures and activities. § 1. Introduction In a recent paper [1] it was shown that the correlation functions of a lattice gas with negative two -body interactions have some cluster property not only at low activity, as known before [2—4] but also for all values of the activity z inside the Lee-Yang circle defined by

A=

Σ

'

ψ(y)

(!)

Similar results [1] have been obtained for purely repulsive potentials but with, the Lee-Yang circle replaced by the circle of convergence of the Mayer series.

Fig. 1 * On leave of absence from Aix-Marseille University. 19 Commun. math. Phys.,Vol. 12

270

G. GALLAVOTTI and S. MIRACLE-SOLE :

In Ref. fl] analyticity properties in β and z of the pressure and the correlation functions are proved for z in any compact not intersecting the Lee-Yang circumference \z = expβA, in the attractive case, or for z in the interior of the circle of convergence of the Mayer series, in the repulsive case. In this paper we deduce results, complementary to the above cited ones, concerning clustering and analyticity at high temperature for lattice gases with very general interactions (not necessarily attractive or repulsive and possibly involving many-body interactions). We show that there exists β0 > 0 such that the Ursell functions u(X) are analytic in z and β for z in an open region G of the complex plane including an open strip around the real positive axis (see Fig. 1 : C has the form of the complementary of the dashed set) and β in a neighborhood Ic of the set |Re/3| < βQ, Imβ = 0. Furthermore if X is a configuration X = {#15 xz, . . .,i%(z)} and X = Σ! \j X2 is any decomposition of X into two configurations at a distance d(Xl9 Xz), then there exist θ > 0, α > 0 such that: exp - α

»

. ,(z,β)ζCχIc,

(2)

where λ ^ -f σo is the range of the interaction. If λ — -f the following weaker result still holds for (z, β) ζ C X Ic : lim

d(Xι,XJ-+ N(XJ + N(X2) fixed

w(Z 1 uZ 2 ) = 0.

(3)

The techniques used to obtain (2) and (3) are similar to the ones used in Ref. [1] except that we obtain bounds on u(X) and analyticity regions by using integral equations instead of the Lee-Yang theorem on attractive potentials or the Groeneveld alternating sign property for positive potentials. We remark that (2) implies that Σ |«(Z)|< + oo. (4) 0€X N(X) fixed

§ 2. The Interaction Potentials Suppose the particles are on a ^-dimensional lattice Zv and interact through symmetric translationally invariant many particle potentials ΦK

^U)) if X = K^2,

,aW

We consider only interactions involving a finite number of particles such that Φ (2> (x, x) = -f- oo and we call Jt the set of the sites occupied by particles in X (so if X Φ -X" the configuration X has zero probability).

Cluster Property Above Critical Temperature

271

Furthermore we suppose finiteness of the energy of the origin, i.e. ||Φ||= Σ

OζX

(5)

\Φ(Σ)\ < + .

x=x We denote by 23 the class of potentials described above. Since we shall be interested in the analyticity properties in the activity z = exp — β Φ ί1) it is useful to write Φ = (Φ^1), Φ') where Φ' is obtained from Φ by setting the one particle potential Φ^1) (which may be interpreted as minus the chemical potential) equal to zero. It is also useful to introduce the potential £?Φ £23 defined, for (X) = (- !)* Σ Φ(S) ,

Φ £93

(6)

(this potential is related to the symmetry properties of the lattice gas under the exchange of particles and holes [5]) and the quantities: A = Σ Φ($) (energy of the origin) , (7)

'« - 1) - 1)] , α = θxp j 8 0 ||Φ'||,

α'^θxpjSolK-SPΦ)!.

( J

(9)

§ 3. Review of Useful Results The following properties are proven in Refs. [2— 6] : i) Consider the two circles centered on the real axis respectively at ~τ~τ37ΐy and z = — α' expβ0A [where α, γ, γ', A are defined in (7), (8), (9)] and with respective radii: / 2\ _ \ \ an(^- j' α/ expj80^4.. Suppose j50 is so small that the maximum real z on the left circumference is, as in Fig. 1, inside the right circle. If C is any closed set in the complementary C of the dashed region drawn in Fig. 1 it is possible to choose β0 small enough that there exists a neighborhood Ic of the set |Re/?| < βQ9 Imβ = 0 such that the correlation functions ρβφ(X) are analytic functions in (β,z) £/ c X C. In this case there also exist [5, 6] a constant Θ0 > 1 such that \ρβ9(X)\ 0, with β0 sufficiently small, and z is both in C and inside the right circle we ha1ve\ρβφ(X)\ ^ θ'0 . However under the same conditions but with z outside the left circle we have, using the symmetry between holes and particles, that Qβφ(X) N(S} ^ Σ (~l) Qβ^φ(X) (for X - 1) and \ρβ